from wiki: https://en.wikipedia.org/wiki/Benacerraf%27s_identification_problem

"In the philosophy of mathematics, Benacerraf's identification problem is a philosophical argument developed by Paul Benacerraf against set-theoretic Platonism and published in 1965 in an article entitled "What Numbers Could Not Be". Historically, the work became a significant catalyst in motivating the development of mathematical structuralism.

The identification problem argues that there exists a fundamental problem in reducing natural numbers to pure sets. Since there exists an infinite number of ways of identifying the natural numbers with pure sets, no particular set-theoretic method can be determined as the "true" reduction. Benacerraf infers that any attempt to make such a choice of reduction immediately results in generating a meta-level, set-theoretic falsehood, namely in relation to other elementarily-equivalent set-theories not identical to the one chosen. The identification problem argues that this creates a fundamental problem for Platonism, which maintains that mathematical objects have a real, abstract existence. Benacerraf's dilemma to Platonic set-theory is arguing that the Platonic attempt to identify the "true" reduction of natural numbers to pure sets, as revealing the intrinsic properties of these abstract mathematical objects, is impossible. As a result, the identification problem ultimately argues that the relation of set theory to natural numbers cannot have an ontologically Platonic nature."

From https://plato.stanford.edu/entries/philosophy-mathematics/#BenEpiPro

"3.4 Benacerraf’s Epistemological Problem
Benacerraf formulated an epistemological problem for a variety of platonistic positions in the philosophy of science (Benacerraf 1973). The argument is specifically directed against accounts of mathematical intuition such as that of Gödel. Benacerraf’s argument starts from the premise that our best theory of knowledge is the causal theory of knowledge. It is then noted that according to platonism, abstract objects are not spatially or temporally localized, whereas flesh and blood mathematicians are spatially and temporally localized. Our best epistemological theory then tells us that knowledge of mathematical entities should result from causal interaction with these entities. But it is difficult to imagine how this could be the case.

"Today few epistemologists hold that the causal theory of knowledge is our best theory of knowledge. But it turns out that Benacerraf’s problem is remarkably robust under variation of epistemological theory. For instance, let us assume for the sake of argument that reliabilism is our best theory of knowledge. Then the problem becomes to explain how we succeed in obtaining reliable beliefs about mathematical entities.
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"4.1 What Numbers Could Not Be
As if saddling platonism with one difficult problem were not enough (section 3.4), Benacerraf formulated a challenge for set-theoretic platonism (Benacerraf 1965). The challenge takes the following form.

There exist infinitely many ways of identifying the natural numbers with pure sets. Let us restrict, without essential loss of generality, our discussion to two such ways:

******There is some math notation I cannot reproduce here. To see it go to the website

I:0123⋮II:0123⋮=∅={∅}={{∅}}={{{∅}}}=∅={∅}={∅,{∅}}={∅,{∅},{∅,{∅}}}
The simple question that Benacerraf asks is:

Which of these consists solely of true identity statements: I or II?

It seems very difficult to answer this question. It is not hard to see how a successor function and addition and multiplication operations can be defined on the number-candidates of I and on the number-candidates of II so that all the arithmetical statements that we take to be true come out true. Indeed, if this is done in the natural way, then we arrive at isomorphic structures (in the set-theoretic sense of the word), and isomorphic structures make the same sentences true (they are elementarily equivalent). It is only when we ask extra-arithmetical questions, such as ‘1∈3?’ that the two accounts of the natural numbers yield diverging answers. So it is impossible that both accounts are correct. According to story I, 3={{{∅}}}, whereas according to story II, 3={∅,{∅},{∅,{∅}}}. If both accounts were correct, then the transitivity of identity would yield a purely set theoretic falsehood.

Summing up, we arrive at the following situation. On the one hand, there appear to be no reasons why one account is superior to the other. On the other hand, the accounts cannot both be correct. This predicament is sometimes called labelled Benacerraf’s identification problem."